[DF] Non-Euclidean Architecture in dungeons questions.

[Edges]

Originally Posted by Lamech

No, a line can't parallel itself.

Form a mobius strip out of paper. Draw a line down the center all the way around and call this line A. Choose a point on line A and call it O. Now draw a ray from O at a very small angle to A. Call this ray B. You will note that as you draw line B it will slowly get farther from A. Once you have gone all the way around so that you are drawing next to O, you will find that you have missed O. But you are now drawing parallel to B. This is what I meant by a line paralleling itself. You can keep going around and laying B down next to itself over and over as it continues to get farther from A and closer to the edge each time.

Now if B is a line and not a ray, then you have the situation of B getting farther from A and closer to the edge in both directions. Now after your first pass, you can hold the paper up to the light and see that B has also intersected itself. This allows for the sort of situation Ulzgoroth was presumably referring to, that of multiple intersections. This occurs not only in the 2-line scenario that he brings up, but is found even in the one line situation.

Originally Posted by Lamech

...its very important to not confuse people who assume that "non-euclidean geometry" means any geometry that is not euclidean.

Yes. This is why I gave the definition of what non-euclidean geometry was. It seemed that you and others were confused (not everyone... e.g. sir_pudding had the right of it). Forgive me if I presumed you were confused when you weren't. It just seemed that way. (In my defense, you did say in post 19 that you didn't know).

But this distinction between not euclidean and non-euclidean isn't trivial. It defines what the thread is about. The OP asked about non-euclidean architecture. To start talking about systems which might violate some postulates but not the 5th gets off topic and runs the risk of confusing people. I would think that bringing up such a system is the perfect time to point out what non-euclidean means.

Originally Posted by Lamech

1)For every point A and for every point B not equal to A there exists a unique
line that passes through A and B.
This fails since on a Mobius strip if you draw a line going along the strip when you get back to the start (what you called a line parallel to itself) you'll start hitting points that you could hit by drawing a different a shorter line.

I'm not sure what you're saying here. But it's true that mobius strips don't follow the first postulate. In a simple mobius strip (i.e. one in which the surface is not non-euclidean and is therefore called euclidean space at every point), there can be more than one line through a given two points. This is because arbitrarily small angles to the edge can be chosen leading to lines that pass themselves arbitrary-many times and because in mobius strips, betweenness isn't strictly defined.

Originally Posted by Lamech

2. For every segment AB and for every segment CD there exists a unique point
E such that B is between A and E and such that segment CD is congruent to
segment BE.
This fails as well. A simple line going along the strip has a maximum length. A line that is 90 degrees to that also would have either a maximum length or a infinite length. Either way it becomes pretty easy to make it so one of the lines can't be extended far enough.

This one doesn't work though. The 2nd postulate can hold for a mobius strip. The second postulate basically says that you can extend any segment by an arbitrarily-large, yet finite amount. It essentially assumes no edge to the plane. While the basic mobius strip that one cuts out of paper has an edge, in topology (and again, a mobius strip is a topological object), it is valid to set boundary points at infinity. One could mathematically construct an infinitely wide mobius strip. Like a klein bottle, it couldn't be embedded in euclidean 3-space without intersections. But that doesn't make it's surface non-euclidean.

What I was ineffectually getting at in my last post was that if it doesn't violate the 5th postulate, it is not non-euclidean.
______________

Basically it seems like we're talking past each other. You seem to be arguing that mobius strips are not euclidean. I conceded this in post 46. And I'm saying they are not required to be non-euclidean. You seem to acknowledge the 5th postulate's importance in your last post. Looking back it appears that we have both been guilty of thinking the other was arguing against something they weren't. Sadly, on my limited time on forums, this sort of thing seems to be the norm.

Regards.